Ionic transport in silica xerogels investigated by dynamic current-voltage characteristics

Ionic transport in silica xerogels investigated

by dynamic current-voltage characteristics

Faculty o f Technical Physics and Applied Mathematics, Gdansk University o f Technology, ul. Gabriela Narutowicza 11/12, 80-952 Gdańsk, Poland.

The essence o f the dynamic current-voltage characteristics (DCVCs) method is the application of linearly changing voltage to a sample with a plane-parallel electrode system. Previous theoretical considerations have shown that the method may be applied to low thermal conductivity materials,

e.g., S i 0 2 porous xerogels, in order to investigate ionic transport therein. The low thermal conductivity o f S i 0 2 xerogels entails a high temperature gradient occurring when DCVCs are measured in vacuum at elevated temperatures. Therefore, it is possible to deduce the sign and mobility o f ions responsible for electrical conductivity inside the materials from the characteristic shape o f DCVCs.

Keywords: porous xerogels, high temperature gradient, dynamic current-voltage characteristics, ionic transport, ionic mobility.

1. Introduction

In recent years, electrical properties of porous solids, in particular those obtained by the sol-gel method [ 1 ]—[5], have been a subject of detailed studies. However, despite numerous attempts, little is known about ionic mobility in such materials [4]. Additionally, the available information concerning the type and sign of the moving ions is still somewhat questionable, because of certain difficulties that usually appear in experimental practice. As ionic mobility in the investigated solids is extremely low, e.g., of 10“12- 10“13 m 2 /Vs at room temperature for silica xerogels [4], measurements of the current are required to last for several hours, which may entail significant dielectric polarization. M oreover, ionic transport in porous materials usually exhibits significant dispersion. As a result, one has to deduce the tim e-of-flight of charge carriers from broadened pulses, strongly disturbed by the polarization current. Another experimental difficulty involves the lack of electrodes effectively injecting ions into an examined solid. Liquid ionic electrodes, sometimes used to investigate polymers ([6], [7]) cannot be applied, as measurements of porous samples must be carried out in vacuum. Additionally, strongly non-uniform temperature distribution within the sample, due to its extremely low heat conductivity, may influence the mobility o f ions

and complicate the analysis of the current traces. These facts show a significant disadvantage o f the conventional tim e-of-flight method.

In the present study, a technique based on dynamic current-voltage characteristics (DCVCs), in which the sample voltage varies continuously at a constant rate, has been applied to investigate the drift mobility of ions in silica xerogels. The method also makes use o f the presence o f a high temperature gradient in the sample, exceeding in our experiment the value of 103 K/cm. Such temperature gradient involves a considerable non-equilibrium dissociation. The dissociation of ion-generating admixtures present in the xerogel is much stronger in the near-electrode area o f higher temperature. Hence, a narrow surface region of the sample o f elevated ion concentration may act as a virtual ionic electrode. W ith a known sign of the electrode potential, mainly ions of the same sign will influence the value and pattern of current in the tested specimen. Theoretical fundamentals and model calculations for DCVCs are presented in [8]. A short comment on the results is given in the next section.

The scope o f this paper is to present a comparison of theoretical presum ptions with DCVCs m easured in doped silica xerogels, S i0 2, and the use o f these results to determine ionic mobility in the investigated material.

2. Theoretical fundamentals

The discussed DCVCs have been determined in the set-up presented in Fig. 1. The principle o f the experiment is to measure the current 7(r), flowing through a specimen of thickness d in the presence of temperature gradient fiT = (Ty-T^ld (with Tx and T2 being the temperatures of the warm and the cold sample surfaces,

respectively) at a linearly increasing field intensity, E(t) = (3Et (with the field increase rate). Neglecting the influence of space charge, carrier diffusion and the

displacement current on the considered course of I(t), the equation o f one-sign-carrier

transport has been solved

E ( t ) ~ [ n ( x ) n ( x , t)l + = 0 (1)

where n(x, t) denotes free carrier concentration. Ionic mobility ji{x) is assumed to

follow the relationship r w i

* x ) = ^ e x p r a J - (2)

In Eqs. (1) and (2), Hq stands for the pre-exponential factor, Wa - for activation energy, k - for the Boltzmann constant, T(x) - for temperature along the sample (see F ig .l).

Having introduced the function

u(x, t) = n(x)n(x, t) (3)

into Eq. (1), one obtains the following formula for the current 7(0, flowing in the specimen:

d

/( t) = ^ | & \ u ( x , t ) d x (4)

0

where e is the elementary charge, and S is the electrode surface. In order to analyze

the experimental data, we shall recall the case o f continuous generation o f electric charge in the thin near-electrode layer of the specimen. According to Eq. (4), the current in the m easuring circuit is given by

7( 0 = SinE (0 * o(0 » (5a)

W = g inE (t)d , t> r (5b)

where g in = efi(0)u(0)Sld is the sample conductance related to the charge generated in

the near-electrode layer, and r - the time of flight o f ions. The function *0(0 describes the instantaneous position o f the carrier front. It can be obtained by means of numerical calculations from the relationship

*o(0

J

(6)

Figure 2 exhibits exemplary DCVCs computed from form ulae (5a) and (5b) for param eters corresponding to the experimental conditions: = 0.34 V/cms, d = 0.05 cm, (3t = 1260 K/cm, T2 - 350 K, n 0 = 2 cm 2/Vs and Wa = 0.65 eV.

Fig. 2. Theoretical DCVCs for the case o f broadening carrier packet moving towards the colder (solid line) and the warmer (dashed line) sample surfaces. The parameters are in the text.

The courses shown in Fig. 2 are related to two ion-generating regions, i.e., to a

layer of either higher Tx or lower T2 temperature (the latter case refers to ionic

injection). A pronounced difference in DCVC courses for time t < x enables us to

determine at which sample surface an excess ion packet is generated. W ith known direction of electric field we also know the sign of ions responsible for the measured DCVCs. As can be seen from the courses, the time of flight of ions is also pronounced: this time being over, the I(t) relationship proceeds from a power into a linear one. From

DCVCs obtained for different tem peratures T2 one can independently determine

activation energy Wa, as well as the jU0 factor, hence also the effective ion mobility [8].

3. Experiment and results

3.1. Details of procedure

S i0 2 silica xerogels obtained by the sol-gel method have been tested. Ionogeneous 0.03 mole RbCl salt was doped into xerogel prior to starting the hydrolysis and polycondensation processes. Details of the procedure to obtain S i0 2 xerogels used in our experiment are described in [9]. Disc-shaped samples of thickness d of ca.

5x10~2 cm were put inside a vacuum test chamber (p ~ 1 Pa), allowing us to stabilize

temperature with an accuracy of AT = ±0.5 K and to independently measure T l and T2.

M icroprocessor-controlled power supply (Stanford Research Systems PS 325/2500V- 25W) made it possible to set the rate of sample voltage variation. During measurements, sample voltage was increased at (3V = j3Ed = 1 V/min. The measuring

process was controlled by a computer program autom atically recording time t, voltage V(t), current I(t) and temperatures Tl and T2.

3.2. Results

Figure 3 displays a typical DCVCs course obtained in the experiment. M easuring points have been obtained for steady tem peratures on both surfaces of the sample:

Fig. 3. Experiment - a comparison with theory. Measuring points have been obtained for 7, = 423 K and T2 = 353.8 K. The fitting parameters: Wa = 0.61 eV, = 2 cm2/V s. The arrow marks the ions time-of-flight r = 4440 s.

to vacuum). The electric field was directed towards the surface of the higher tem perature T{. By comparison of curves in Figs. 3 and 2, one may see that the current

derives from carriers being generated in the near-electrode region of temperature Tx.

Taking into account the direction o f the electric field, one may conclude that measured current is mainly due to negative ion packet drift. In Fig. 3, the arrival time r o f an ion packet front to the opposite electrode o f tem perature T2 is marked. With a known

time t, a com puter program searching for the best fit o f the theoretical curve allows us to find an adequate approximation o f the activation energy Wa and the

pre-exponential factor /i 0. W ith given values of Wa and ju0, it is possible to determine

ionic mobility pi, which - d u e to a high temperature gradient in the sam ple-

corresponds well to temperature T2. M aking use of the expression E(t) = fiEt and of

Eq. (6), one may obtain

(7)

For a high tem perature gradient, the main contribution to the integral in Eq. (7) is made by the near-electrode region of temperature T2. Hence, the formula (7) may be

approximately replaced with

2 2 * ^ f

T

~^PrM

0

Wa

CXPU

Equation (8) allows us to determine the mean values o f energy Wa and factor ju0

from the experimental relationship t = f ( T 2). The value of Wa may be read from the

slope, the value of jU0 - from abcissa of the plotted line lo g(t2(3t/ T 2) = / ( 103/ r 2).

Using the least squares method, activation energy Wa = 0.65 eV and factor

Fig. 4. Experimentally determined ions time-of-flights r = f(T2) (full circles) and the fitting line

\og(v2 /3t/T^) = / ( 1 0 3/ r 2). From its slope and abcissa the mean values o f Wa = 0.65 eV and ^0 = 7.16 cm2/V s have been obtained. The correlation coefficient R2 is also indicated.

Fig. 5. Exponential temperature dependence o f negative ion mobility in S i 0 2 silica xerogel with 0.03 mole RbCl admixture. The curve has been obtained for parameters Wa and Hq resulting from Fig. 4.

Finally, from Eq. (2) and the already determ ined param eters Wa and /Iq, we may establish the tem perature relationship of negative ion m obility for the examined xerogel in the test tem perature range of 330-370 K. The relationship is shown in Fig. 5.

4. Conclusions

The paper deals with preliminary results of tests limited to negative ions. Their transport in S i0 2 xerogels has been investigated in a relatively narrow tem perature range with a dynamic electric field. Despite these lim itations, the presented results have confirmed the theoretically predicted possibility to use DCVCs courses to investigate ionic transport in porous S i0 2 xerogels. Extremely low heat conductivity of the material involves a high temperature gradient in the samples and results in the characteristic shape of DCVCs. The relationship between DCVC shape and the direction of ionic drift (in accordance with or opposite to the temperature gradient),

as well as clearly pronounced times o f flight, allow us to determine the sign o f ions, their mobility and activation energy. The suggested method of determining these values appears to open a way to a more accurate investigation o f ionic transport mechanism in porous materials.

References

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